Exercise 4.4: Details of Symmetrisation Argument
(a)
For any \(g \in \mathscr{G}\), \(g(X) \leq |g(X)|\) a.s. Taking expectation of both sides \begin{align} \E [g(X)] \leq \E [\sup_{g \in \mathscr{G}} |g(X)|] \, . \end{align} Because this holds for all \(g \in \mathscr{G}\), we can conclude by taking the supremum over the left-hand side.
To prove the inequality in equation (4.17), use the Jensen’s inequality to pull the expectation over \(Y\) outside of the absolute value, and then apply the above result.
(b)
For any \(g \in \mathscr{G}\), \(\Phi(\E|g(X)|) \leq \Phi(\E [\sup_{g \in \mathscr{G}} |g(X)|])\) because \(\Phi\) is increasing. Since \(\Phi\) is also convex, the result follows by the Jensen’s inequality.
In the proof of Proposition 11, use the Jensen’s inequality to pull the expectation over \(Y\) outside of the absolute value again.